A053871 -id:A053871 - cp's OEIS frontend

A055142 Expansion of e.g.f.: exp(x)*sqrt(1-2x).

1, 0, -2, -8, -36, -224, -1880, -19872, -251888, -3712256, -62286624, -1171487360, -24402416192, -557542291968, -13861636770176, -372514645389824, -10759590258589440, -332386419622387712

Offset: 0

Christian G. Bower, May 09 2000

sign

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001147, A053871.

Column 0 of triangle A055141.

CoefficientList[Series[E^x*Sqrt[1-2*x], {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Mar 29 2014 *)

def A055142(n):
    @CachedFunction
    def h(n):
        return (-1)^n*2*(n-2)*abs(h(n-1)+h(n-2)) if n>1 else 1
    return -(-1)^(n+1)*h(n+1)
[A055142(n) for n in (0..17)]  # Peter Luschny, Nov 02 2012

a(n) ~ -2^(n-1/2) * n^(n-1) / exp(n-1/2). - Vaclav Kotesovec, Mar 29 2014

D-finite with recurrence: a(n) +2*(-n+1)*a(n-1) +2*(n-1)*a(n-2)=0. - R. J. Mathar, Jan 22 2020

A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

Original entry on oeis.org

1, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 2, 0, 0, 1, 0, 9, 8, 0, 0, 1, 0, 34, 252, 60, 0, 0, 1, 0, 125, 5672, 14337, 544, 0, 0, 1, 0, 461, 125750, 2604732, 1327104, 6040, 0, 0, 1, 0, 1715, 2857472, 488360625, 2533087904, 182407545, 79008, 0, 0

Offset: 0

Alois P. Heinz, Feb 16 2024

			A(2,3) = 9: 124|356, 125|346, 126|345, 134|256, 135|246, 136|245, 145|236, 146|235, 156|234.
Square array A(n,k) begins:
  1, 1,   1,       1,          1,             1, ...
  0, 0,   0,       0,          0,             0, ...
  0, 0,   2,       9,         34,           125, ...
  0, 0,   8,     252,       5672,        125750, ...
  0, 0,  60,   14337,    2604732,     488360625, ...
  0, 0, 544, 1327104, 2533087904, 5192229797500, ...

Alois P. Heinz, Antidiagonals n = 0..54, flattened
Wikipedia, Partition of a set

Columns k=0+1,2-3 give: A000007, A053871, A370357.
Rows n=0-2 give: A000012, A000004, A010763(n-1) for k>0.
Main diagonal gives A370367.
Antidiagonal sums give A370368.
Cf. A060540, A370363.

A:= proc(n, k) `if`(k=0,`if`(n=0, 1, 0), add(
      (-1)^(n-j)*binomial(n, j)*(k*j)!/(j!*k!^j), j=0..n))
    end:
seq(seq(A(n, d-n), n=0..d), d=0..10);

A(n,k) = A060540(n,k) - A370363(n,k) for n,k >= 1.

A088992 Derangement numbers d(n,5) where d(n,k) = k(n-1)(d(n-1,k) + d(n-2,k)), with d(0,k) = 1 and d(1,k) = 0.

Original entry on oeis.org

1, 0, 5, 50, 825, 17500, 458125, 14268750, 515440625, 21188375000, 976671703125, 49893003906250, 2797832158515625, 170863509745312500, 11287987223748828125, 802119551344589843750, 61005565392625400390625, 4944614795517599218750000

Offset: 0

N. J. A. Sloane, Nov 02 2003

In general, d(n,k) is asymptotic to sqrt(2*Pi) * k^n * n^(n + 1/2) / (Gamma(1/k) * exp((n*k+1)/k) * n^((k-1)/k)), for k>0. - Vaclav Kotesovec, Oct 31 2017

Cf. A000166, A053871, A033030, A088991.

Inverse binomial transform of A008548. E.g.f.: exp(-x)/(1-5*x)^(1/5). - Vladeta Jovovic, Dec 17 2003
a(n) ~ Pi * sqrt(2) * n^(n-3/10) * 5^n / (sqrt(Pi) * Gamma(1/5) * exp(n + 1/5)). - Vaclav Kotesovec, Oct 31 2017
a(n) = (-1)^n * n! * Sum_{k=0..n} 5^k * binomial(-1/5,k)/(n-k)!. - Seiichi Manyama, Apr 23 2025

A165968 Number of pairings disjoint to a given pairing, and containing a given pair not in the given pairing.

Original entry on oeis.org

0, 1, 2, 10, 68, 604, 6584, 85048, 1269680, 21505552, 407414816, 8535396256, 195927013952, 4890027052480, 131842951699328, 3818743350945664, 118253903175951104, 3898687202158805248, 136339489775029813760, 5040776996774472673792

Offset: 1

Lewis Mammel (l_mammel(AT)att.net), Oct 02 2009

nonn

The formula is derived by an application of the principle of inclusion and exclusion.
In reference to A053871, it is observed that the set of pairings disjoint to a given pairing can be partitioned into 2n-2 equivalent sets according to the 2n-2 pairs containing a given item. So it is seen that each term of that sequence must be divisible by 2n-2, giving the corresponding term of this sequence. However, the formula given here is derived independently.
Hankel transform of a(n+1) is A168467. Binomial transform of a(n+1) is A001147(n+1). - Paul Barry, Jan 26 2011
a(n) is a subset of the set of pairings of the first 2n integers (A001147) in another way: forbidding pairs of the form (2k,2k+1) for all k. - Olivier Gérard, Feb 08 2011

			a(1) = 0 trivially.
a(2) = 1 since there is a unique pairing disjoint to the canonical pairing, 01 23, and containing any of the 4 pairs not in the canonical pairing.
a(3) = 2 since there are 2 pairings disjoint to the canonical pairing, 01 23 45, and containing the pair 02, not in the canonical pairing: 02 14 35 and 02 15 34.

John Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, Chapter 3

Michael De Vlieger, Table of n, a(n) for n = 1..405
Jean-Luc Baril, Avoiding patterns in irreducible permutations, Discrete Mathematics and Theoretical Computer Science, Vol 17, No 3 (2016).
Marilena Barnabei, Niccolò Castronuovo, and Matteo Silimbani, Hertzsprung patterns on involutions, arXiv:2412.03449 [math.CO], 2024. See p. 10.
Célia Biane, Greg Hampikian, Sergey Kirgizov, and Khaydar Nurligareev, Endhered patterns in matchings and RNA, arXiv:2404.18802 [math.CO], 2024. See pp. 8-9.

Cf. A001147, the double factorial.
Cf. also A053871.
Cf. A168467.

a:= n-> add((-1)^(n-k-1)*binomial(n-1,k)*(2*(k+1))!/(2^(k+1)*(k+1)!), k=0..n-1):
seq(a(n), n=0..20);

a[n_] := Sum[(-1)^(n-k-2)* Binomial[n-2, k]*(2*(k+1))!/(2^(k+1)*(k+1)!) , {k, 0, n-2}]; a /@ Range[20]
(* Jean-François Alcover, Jul 11 2011, after Maple *)
CoefficientList[Series[-1+1/(E^x*Sqrt[1-2*x]) + Sqrt[2]*DawsonF[1/Sqrt[2]] + Sqrt[-Pi/(2*E)]*Erf[Sqrt[-1/2+x]],{x,0,20}],x]*Range[0,20]! (* Vaclav Kotesovec, Feb 04 2014 *)

define a(n)
{
    auto sign, i,s;
    s=0; sign = 1;
    for ( i=0 ; i<=n-1 ; i++ ) {
        s = s + sign * ffac(n-1-i) * c( n-2, i );
        sign = sign * -1;
    }
    return s;
}
/* returns (2n-1)!! */
define ffac( n )
{
    if ( n <= 1 ) return 1;
    return  (2*n-1)* ffac(n-1);
}
/* returns combinations of n things taken i at a time */
define c(n,i)
{
    auto j,s;
    s=1;
    if ( n < 0 ) return 0;
    for ( j=0 ; j

Formula

a(n) = (2n-3)!! - C(n-2,1) * (2n-5)!! + ... +/- C(n-2,n-1)*3!! -/+ 1.

a(n) = (2*n-4)*a(n-1) +(2*n-6)*a(n-2) for n>2. - Gary Detlefs, Jul 11 2010

G.f.: x/(1-2x/(1-3x/(1+x-4x/(1-5x/(1+x-6x/(1-7x/(1+x-8x/(1-... (continued fraction). - Paul Barry, Jan 26 2011

a(n) = Sum_{k=0..n-1} (-1)^(n-k-1)*C(n-1,k)*(2*(k+1))!/(2^(k+1)*(k+1)!). - Paul Barry, Jan 26 2011

Conjecture: a(n) +2*(-n+1)*a(n-1) +2*(-n+2)*a(n-2)=0. - R. J. Mathar, Nov 15 2012

G.f.: x/G(0) where G(k) = 1 - 2*x*(2*k+1) - x^2*(2*k+2)*(2*k+3)/G(k+1) (continued fraction). - Sergei N. Gladkovskii, Jan 13 2013

G.f.: Q(0)-1, where Q(k) = 1 - x*(k+1)/( x*(k+1) - (1 +x)/Q(k+1) ); (continued fraction). - Sergei N. Gladkovskii, Oct 21 2013

a(n) ~ 2^(n-1/2) * n^(n-1) / exp(n+1/2). - Vaclav Kotesovec, Feb 04 2014

a(n) = A053871(n)/(2n-2) for n>1.

a(n) * (2n-2) satisfies the recurrence of A053871, so Detlef's conjecture was correct. And if we rewrite Mathar's conjecture as b(n) = 2*(n-1)*b(n-1) +2*(n-2)*b(n-2) it becomes quite clear that Mathar's b(n) = a(n-1). - Sergey Kirgizov, Jun 03 2022

A289191 Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side and non-isomorphic under rotational symmetry.

Original entry on oeis.org

0, 2, 4, 22, 112, 1060, 11292, 149448, 2257288, 38720728, 740754220, 15648468804, 361711410384, 9081485302372, 246106843197984, 7160143986526240, 222595582448849152, 7364186944683168828, 258327454310582805036, 9577476294162996275928, 374205233351106756670120

Offset: 1

Marko Riedel, Jun 27 2017

nonn

The case n=2 is a degenerate polygon (two sides connecting two vertices). The two possibilities are when the edges cross and do not cross. Polygons start at n=3 with a triangle.

The sequence A132102 enumerates the case that edges are allowed between exits on the same side. This sequence can be enumerated in a similar manner using inclusion-exclusion on the number of sides that have their two exits connected. - Andrew Howroyd, Jan 26 2020

Andrew Howroyd, Table of n, a(n) for n = 1..200
Marko Riedel, Hexagonal tiles.
Marko Riedel, Maple code to compute number of tiles by ordinary enumeration and by Power Group Enumeration.
Marko Riedel, Maple code for number of tiles using Burnside lemma.

See A053871 for tiles with no rotational symmetries being taken into account, A289269 for tiles with rotational and reflectional symmetries being taken into account, A289343 for the same statistic evaluated when n is prime.

Cf. A132102.

a(n) = {sumdiv(n, d, my(m=n/d); eulerphi(d)*sum(i=0, m, (-1)^i * binomial(m, i) * sum(j=0, m-i, (d%2==0 || m-i-j==0) * binomial(2*(m-i), 2*j) * d^j * (2*j)! / (j!*2^j) )))/n} \\ Andrew Howroyd, Jan 26 2020

Terms a(14) and beyond from Andrew Howroyd, Jan 26 2020

A064280 Number of nonequivalent solutions to the order n checkerboard problem up to reflection and rotation: place n pieces on an n X n board so there is exactly one piece in each row, column and main diagonal.

Original entry on oeis.org

1, 0, 0, 1, 4, 12, 86, 696, 6150, 61760, 673256, 8137200, 105074420, 1479237312, 22077680616, 354753059584, 6007578698408, 108500041654272, 2055204828592832, 41215470268919040, 863378484993573840, 19036646809582054400, 436944006380312366240

Offset: 1

Jud McCranie, Sep 24 2001

nonn

For even n>=4: A007016(n) = 8*A064280(n).

For even n, the diagonals do not intersect and there can be no symmetrical solutions. For odd n, a symmetrical solution will have a rook on the central square and the remaining n-1 rooks must be placed so as to avoid the main diagonals. See A292080 for information on counting non-attacking rook configurations with no rook on either main diagonal. - Andrew Howroyd, Sep 12 2017

			The 4 X 4 solution is unique, up to equivalence, with pieces at (1,1), (2,3), (3,4) and (4,2).

Andrew Howroyd, Table of n, a(n) for n = 1..100
Geoffrey Chase, Checkerboard Problem Solved, Creative Computing 6(1), Jan 1980, 122.
Bahairiv Joshi, Unique Solutions to the Checkerboard Problem, Creative Computing 6(10), Oct 1980, 124-125.
Abijah Reed, Comments on Checkerboard Problem Solved, Creative Computing 6(5), May 1980, 94.

A007016 gives the number of solutions including symmetrical ones.

Cf. A000166, A007016, A037224, A053871, A292080.

sf = Subfactorial;
x[n_] := x[n] = Integrate[If[EvenQ[n], (x^2 - 4*x + 2)^(n/2), (x - 1)*(x^2 - 4*x + 2)^((n - 1)/2)]/E^x, {x, 0, Infinity}];
F[n_ /; EvenQ[n]] := With[{m = n/2}, m*(x[2*m] - (2*m - 3)*x[2*m - 1])];
F[n_ /; OddQ[n]] := With[{m = (n - 1)/2}, (2*m + 1)*x[2*m] + 3*m*x[2*m - 1] - 2*m*(m - 1)*x[2*m - 2]];
d[n_] := (-1)^n HypergeometricPFQ[{1/2, -n}, {}, 2];
R[n_] := If[OddQ[n], 0, If[n == 0, 1, (n - 1)!*2/(n/2 - 1)!]];
a[1] = 1; a[n_] := With[{m = Quotient[n, 2]}, (F[n] + If[EvenQ[n], 0, 2^m * sf[m] + 2*R[m] + 2*d[m] + 2*Boole[m == 0]])/8];
Array[a, 30] (* Jean-François Alcover, Sep 15 2019 *)

\\ here sf is A000166, F is A007016, D is A053871, R(n) is A037224(2n).
sf(n) = {n! * polcoeff( exp(-x + x * O(x^n)) / (1 - x), n)}
F(n) = {my(v = vector(n)); for(n=4, length(v), v[n] = (n-1)*v[n-1] + 2*if(n%2==1, (n-1)*v[n-2], (n-2)*if(n==4,1,v[n-4]))); if(n<4, [1,0,0][n], if(n%2==0, n*(v[n] - (n-3)*v[n-1]), 2*n*v[n-1] + 3*(n-1)*v[n-2] - (n-1)*(n-3)*v[n-3])/2)}
D(n) = {sum(k=0, n, (-1)^(n-k) * binomial(n,k) * (2*k)!/(2^k*k!))}
R(n) = {if(n%2==1, 0, if(n==0, 1, (n-1)!*2/(n/2-1)!))}
a(n) = {(F(n) + if(n%2==0, 0, my(m=n\2); 2^m * sf(m) + 2*R(m) + 2*D(m) + 2*(m==0)))/8} \\ Andrew Howroyd, Sep 12 2017

a(2n) = A007016(2n) / 8, a(2n+1) = (A007016(2n+1) + 2^n * A000166(n) + 2*A037224(2*n) + 2*A053871(n)) / 8 for n > 0. - Andrew Howroyd, Sep 12 2017

a(11)-a(12) from Lars Blomberg, Jan 13 2013

Name clarified and terms a(13) and beyond from Andrew Howroyd, Sep 12 2017

A289269 Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side and non-isomorphic under rotational and reflectional, i.e. dihedral, symmetry.

Original entry on oeis.org

0, 2, 4, 19, 80, 638, 6054, 76692, 1137284, 19405244, 370597430, 7825459362, 180862277352, 4540781512946, 123053646087312, 3580073396748560, 111297799861936256, 3682093529146577694, 129163727524848878358, 4788738149626920381804, 187102616692953377567060

Offset: 1

Marko Riedel, Jun 29 2017

nonn

The case n=2 is a degenerate polygon (two sides connecting two vertices). The two possibilities are when the edges cross and do not cross. Polygons start at n=3 with a triangle.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Marko Riedel, Hexagonal tiles
Marko Riedel, Maple code to compute number of tiles by ordinary enumeration and by Power Group Enumeration
Marko Riedel, Maple code for number of tiles using Burnside lemma.

See A053871 for tiles with no symmetries being taken into account, A289191 for tiles with rotational symmetries only being taken into account.

\\ here R(n) is A289191.
S(n)={sum(i=0, n\2, (-1)^i * sum(j=0, (n-2*i)\2, (2*j)!/j! * if(n%2, if(j, 2*binomial(n\2, i)*binomial(n-2*i-1, 2*j-1)), binomial(n/2, i)*binomial(n-2*i, 2*j) + if(j, binomial(n/2-1, i)*binomial(n-2*i-2, 2*j-2))) / 2))}
R(n)={sumdiv(n, d, my(m=n/d); eulerphi(d)*sum(i=0, m, (-1)^i * binomial(m, i) * sum(j=0, m-i, (d%2==0 || m-i-j==0) * binomial(2*(m-i), 2*j) * d^j * (2*j)! / (j!*2^j) )))/n}
a(n)={(R(n) + S(n))/2} \\ Andrew Howroyd, Jan 26 2020

Terms a(14) and beyond from Andrew Howroyd, Jan 26 2020

A054479 Number of sets of cycle graphs of 2n nodes where the 2-colored edges alternate colors.

Original entry on oeis.org

1, 0, 6, 120, 6300, 514080, 62785800, 10676746080, 2413521910800, 700039083744000, 253445583029839200, 112033456760809584000, 59382041886244720843200, 37175286835046004765120000, 27139206193305890195912400000, 22852066417535931447551359680000

Offset: 0

Christian G. Bower, Mar 29 2000

nonn

Also number of permutations in the symmetric group S_2n in which cycle lengths are even and greater than 2, cf. A130915. - Vladeta Jovovic, Aug 25 2007

a(n) is also the number of ordered pairs of disjoint perfect matchings in the complete graph on 2n vertices. The sequence A006712 is the number of ordered triples of perfect matchings. - Matt Larson, Jul 23 2016

Vincenzo Librandi, Table of n, a(n) for n = 0..200

Cf. A001147, A001818, A053871, A006712.

b:= proc(n) option remember; `if`(n=0, 1, add(
      b(n-2*j)*binomial(n-1, 2*j-1)*(2*j-1)!, j=2..n/2))
    end:
a:= n-> b(2*n):
seq(a(n), n=0..15);  # Alois P. Heinz, Mar 06 2023

Table[(n-1)*(2*n)!^2 * HypergeometricPFQ[{2-n},{3/2-n},-1/2] / (4^n*(n-1/2)*(n!)^2), {n, 0, 20}] (* Vaclav Kotesovec, Mar 29 2014 after Mark van Hoeij *)

x='x+O('x^66); v=Vec(serlaplace(1/(sqrt(exp(x^2)*(1-x^2))))); vector(#v\2,n,v[2*n-1]) \\ Joerg Arndt, May 13 2013

If b(2n)=a(n) then e.g.f. of b is 1/(sqrt(exp(x^2)*(1-x^2))).
a(n) = 4^n*(n-1)*gamma(n+1/2)^2*hypergeom([2-n],[3/2-n],-1/2)/(Pi*(n-1/2)). - Mark van Hoeij, May 13 2013
a(n) ~ 2^(2*n+1) * n^(2*n) / exp(2*n+1/2). - Vaclav Kotesovec, Mar 29 2014

A055141 Matrix inverse of triangle A055140.

Original entry on oeis.org

1, 0, 1, -2, 0, 1, -8, -6, 0, 1, -36, -32, -12, 0, 1, -224, -180, -80, -20, 0, 1, -1880, -1344, -540, -160, -30, 0, 1, -19872, -13160, -4704, -1260, -280, -42, 0, 1, -251888, -158976, -52640, -12544, -2520, -448, -56, 0, 1, -3712256, -2266992

Offset: 0

Christian G. Bower, May 09 2000

T is an example of the group of matrices outlined in the table in A132382--the associated matrix for aC(1,1). The e.g.f. for the row polynomials is exp(x*t) * exp(x) * (1-2*x)^(1/2). T(n,k) = Binomial(n,k)* s(n-k) where s = A055142 with an e.g.f. of exp(x) * (1-2*x)^(1/2) which is the reciprocal of the e.g.f. of A053871. The row polynomials form an Appell sequence. [From Tom Copeland, Sep 11 2008]

			1; 0,1; -2,0,1; -8,-6,0,1; -36,-32,-12,0,1; ...

a(n, k) = A053142(n-k)*C(n, k).

A155517 Triangle read by rows: T(n,k) is the number of permutations p of {1,2,...,n} for which the number of j < ceiling(n/2) such that p(j) + p(n+1-j) = n+1 is equal to k (n>=1; 0<=k <=ceiling(n/2)).

Original entry on oeis.org

0, 1, 0, 2, 4, 0, 2, 16, 0, 8, 64, 48, 0, 8, 384, 288, 0, 48, 2880, 1536, 576, 0, 48, 23040, 12288, 4608, 0, 384, 208896, 115200, 30720, 7680, 0, 384, 2088960, 1152000, 307200, 76800, 0, 3840, 23193600, 12533760, 3456000, 614400, 115200, 0, 3840, 278323200

Offset: 1

Emeric Deutsch, Jan 26 2009

For the permutation 31756284 of S_8 we have k=2 because p(2) + p(7) = 1+8 = 9 and p(3) + p(6) = 7+2 = 9; for the permutation 3214756 of S_7 we have k=2 because p(3) + p(5) = 1+7 = 8 and p(4) + p(4) = 4+4 = 8.
Row sums are the factorial numbers (A000142).
Row n contains 1 + ceiling(n/2)entries.
T(2n,n) = n!*2^n = A037223(2n) = number of centrosymmetric permutations in S[2n];
T(2n+1,n+1) = n!*2^n = A037223(2n+1) = number of centrosymmetric permutations in S[2n+1].
T(n,0) = A155518(n).
Sum_{k=0..ceiling(n/2)} k*T(n,k) = A155519(n).

			T(4,2)=8 because we have 1234, 4231, 1324, 4321, 2143, 3142, 2413 and 3412.
Triangle starts:
    0,   1;
    0,   2;
    4,   0,   2;
   16,   0,   8;
   64,  48,   0,   8;
  384, 288,   0,  48;

Cf. A000142, A037223, A055140, A155518, A155519.

g[0] := 1: g[1] := 0: for n from 2 to 20 do g[n] := (2*(n-1))*(g[n-1]+g[n-2]) end do: T := proc (n, k) if `mod`(n, 2) = 0 then 2^((1/2)*n)*factorial((1/2)*n)*g[(1/2)*n-k]*binomial((1/2)*n, k) else 2^((1/2)*n-1/2)*factorial((1/2)*n-1/2)*g[(1/2)*n+1/2-k]*binomial((1/2)*n+1/2, k) end if end proc: for n to 12 do seq(T(n, k), k = 0 .. ceil((1/2)*n)) end do;

T(2n,k) = n!*2^n*A055140(n,k);
T(2n-1,k) = (n-1)!*2^(n-1)*A055140(n,k);
here A055140(n,k) = A053871(n-k)*binomial(n,k), where g(n) = A053871(n) is defined by g(0)=1, g(1)=0, g(n) = 2(n-1)(g(n-1)+g(n-2)).

cp's OEIS Frontend

A055142 Expansion of e.g.f.: exp(x)*sqrt(1-2x).

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A370366 Number A(n,k) of partitions of [k*n] into n sets of size k having no set of consecutive numbers whose maximum (if k>0) is a multiple of k; square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A088992 Derangement numbers d(n,5) where d(n,k) = k(n-1)(d(n-1,k) + d(n-2,k)), with d(0,k) = 1 and d(1,k) = 0.

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A165968 Number of pairings disjoint to a given pairing, and containing a given pair not in the given pairing.

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Maple

Mathematica

bc

Formula

A289191 Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side and non-isomorphic under rotational symmetry.

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PARI

Extensions

A064280 Number of nonequivalent solutions to the order n checkerboard problem up to reflection and rotation: place n pieces on an n X n board so there is exactly one piece in each row, column and main diagonal.

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Mathematica

PARI

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A289269 Number of polygonal tiles with n sides with two exits per side and n edges connecting pairs of exits, with no edges between exits on the same side and non-isomorphic under rotational and reflectional, i.e. dihedral, symmetry.

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A054479 Number of sets of cycle graphs of 2n nodes where the 2-colored edges alternate colors.

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